PLASTIC BENDING OF BEAMS

Let us consider a beam of homogenous material and symmetrical section subjected to a bending moment M. The distribution of bending stress flows a linear law with zero stress at the neutral axis and a maximum stress at the outermost fibres, when the deformations are within the elastic limit. In case the magnitude of M increases, the stress distribution also changes. These are shown in the following stages.

Figure 1. Stages of Stress Distribution

Stage 1. The deformation is within the elastic limit. The maximum bending stress is f. If the section modulus is Z, we have M = f.Z. (Figure 1(a)).

Stage 2. If the bending moment is gradually increased so that the extreme fibre reaches the yield stress f_y, the corresponding bending moment is given by, M_y = f_y.Z. (Figure 1(b)).

Stage 3. The bending moment, if further increased, will not increase the maximum fibre stress which remains at the yield stress value f_y, but the yield will spread into fibres for a depth e called the depth of penetration. (Figure 1(c)).

Stage 4. If the bending moment is further increased, a stage will be reached when the yield will spread into all the fibres resulting in a stress diagram shown in figure 1(d).

The beam section in this stage has reached its maximum resisting capacity. Any further increase in the bending moment cannot be resisted by the section and an instability is reached as would happen if a hinge was provided at the section. We say that a plastic hinge has formed at the section.

At this stage area of the compression or tension zone of the section equals A/2, where A is the cross sectional area of the beam.

Total compression on the section = Total tension on the section = f_y.A/2

This means, the neutral axis at this stage is called the plastic moment of resistance (or plastic moment) denoted by M_p and is given by

M_p          = Moment of total compression about the plastic neutral axis + Moment of total tension about the plastic neutral axis.

= f_y.A/2 x Distance of the centroid of the compression zone from the plastic neutral axis +   

   f_y.A/2 x Distance of the centroid of the tension zone from the plastic neutral axis.

= f_y.[Sum of the moments of the compression and tension zones about the plastic neutral  

   axis]

= f_y.Z_p

Where Z_p = Sum of the moments of the compression and tension zones about the plastic neutral axis and is called the plastic modulus.

Thus when a beam section develops a plastic hinge,

Plastic moment of resistance = M_p = f_y.Z_p

We know, the ratio of the moment of inertia of the beam section about the elastic i.e., the centroidal neutral axis to the distance of the most distant edge of the section is the section modulus Z of the beam section.

The ratio of the plastic modulus Z_p to the section modulus Z is called the shape factor or form factor denoted by K_s of the section. This is a measure of the reserve strength the section possesses after the initial yielding.

If M_y = Moment of resistance of the section when the most extreme fibre of the section reaches the yield stress f_y,

M_y = f_y.Z